Russian Reports He Has Solved a Celebrated Math Problem

By SARA ROBINSON (NYT) 1310 words

A Russian mathematician is reporting that he has proved the Poincaré Conjecture,
one of the most famous unsolved problems in mathematics.

The mathematician, Dr. Grigori Perelman of the Steklov Institute of Mathematics
of the Russian Academy of Sciences in St. Petersburg, is describing his work
in a series of papers, not yet completed.

It will be months
before the proof can be thoroughly checked. But if true, it will verify a
statement about three-dimensional objects that has haunted mathematicians
for nearly a century, and its consequences will reverberate through geometry
and physics.

If his proof is accepted for publication in a refereed
research journal and survives two years of scrutiny, Dr. Perelman could be
eligible for a $1 million prize sponsored by the Clay Mathematics Institute
in Cambridge, Mass., for solving what the institute identifies as one of
the seven most important unsolved mathematics problems of the millennium.

Rumors about Dr. Perelman's work have been circulating since November,
when he posted the first of his papers reporting the result on an Internet
preprint server.

Last week at the Massachusetts Institute of Technology,
he gave his first formal lectures on his work to a packed auditorium. Dr.
Perelman will give another lecture series at the State University of New
York at Stony Brook starting on Monday.

Dr. Perelman declined to be interviewed, saying publicity would be premature.

For two months, Dr. Tomasz S. Mrowka, a mathematician at M.I.T., has been
attending a seminar on Dr. Perelman's work, which relies on ideas pioneered
by another mathematician, Richard Hamilton. So far, Dr. Mrowka said, every
time someone brings up an issue or objection, Dr. Perelman has a clear and
succinct response.

''It's not certain, but we're taking it very seriously,''
Dr. Mrowka said. ''He's obviously thought about this stuff very hard for
a long time, and it will be very hard to find any mistakes.''

Formulated
by the French mathematician Henri Poincaré in 1904, the Poincaré Conjecture
is a central question in topology, the study of the geometrical properties
of objects that do not change when the object is stretched, twisted or shrunk.

The hollow shell of the surface of the earth is what topologists
would call a two-dimensional sphere. It has the property that every lasso
of string encircling it can be pulled tight to one spot.

On the surface
of a doughnut, by contrast, a lasso passing through the hole in the center
cannot be shrunk to a point without cutting through the surface.

Since the 19th century, mathematicians have known that the sphere is the
only bounded two-dimensional space with this property, but what about higher
dimensions?

The Poincaré Conjecture makes a corresponding statement
about the three-dimensional sphere, a concept that is a stretch for the nonmathematician
to visualize. It says, essentially, that the three-dimensional sphere is
the only bounded three-dimensional space with no holes.

''The hard
part is how to tell globally what a space looks like when you can only see
a little piece of it at a time,'' said Dr. Benson Farb, a professor of mathematics
at the University of Chicago. ''It was pretty reasonable to think the earth
was flat.''

That conjecture is notorious for the many ''solutions''
that later proved false. Indeed, Poincaré himself demonstrated that his earliest
version of his conjecture was wrong. Since then, dozens of mathematicians
have asserted that they had proofs until experts found fatal flaws.

Although many experts say they are excited and hopeful about Dr. Perelman's
effort, they also urge caution, noting that not all of the proof has been
written down and that even the most reliable researchers make mistakes.

That was the case in 1993 with Dr. Andrew J. Wiles, the Princeton professor
whose celebrated proof for Fermat's Last Theorem turned out to have a serious
gap that was repaired after months of effort by Dr. Wiles and a former student,
Dr. Richard Taylor.

Dr. Perelman's results go well beyond a solution
to the problem at hand, as did those of Dr. Wiles. Dr. Perelman's results
say he has proved a much broader conjecture about the geometry of three-dimensional
spaces made in the 1970's. The Poincaré Conjecture is but a small part of
that.

Dr. Perelman's personal story has parallels to that of Dr.
Wiles, who, without confiding in his colleagues, worked alone in his attic
on Fermat's Last Theorem. Though his early work has earned him a reputation
as a brilliant mathematician, Dr. Perelman spent the last eight years sequestered
in Russia, not publishing.

In his paper posted in November, Dr. Perelman,
now in his late 30's, thanks the Courant Institute at New York University,
SUNY Stony Brook and the University of California at Berkeley, because his
savings from visiting positions at those institutions helped support him
in Russia.

His papers say that he has proved what is known as the
Geometrization Conjecture, a complete characterization of the geometry of
three-dimensional spaces.

Since the 19th century, mathematicians
have known that a type of two-dimensional space called a manifold can be
given a rigid geometric structure that looks the same everywhere. Mathematicians
could list all the possible shapes for two-dimensional manifolds and explain
how a creature living on the surface of one can tell what kind of space he
is on.

In the 1950's, however, a Russian mathematician proved that
the problem was impossible to resolve in four dimensions and that even for
three dimensions, the question looked hopelessly complex.

In the
early 1970's, Dr. William P. Thurston, a professor at the University of California
at Davis, conjectured that three-dimensional manifolds are composed of many
homogeneous pieces that can be put together only in prescribed ways and proved
that in many cases his conjecture was correct. Dr. Thurston won a Fields
Medal, the highest honor in mathematics, for his work.

Dr. Perelman's
work, if correct, would provide the final piece of a complete description
of the structure of three-dimensional manifolds and, almost as an afterthought,
would resolve Poincaré's famous question. Dr. Perelman's approach uses a
technique known as the Ricci flow, devised by Dr. Hamilton, who is now at
Columbia University.

The Ricci flow is an averaging process used
to smooth out the bumps of a manifold and make it look more uniform. Dr.
Hamilton uses the Ricci flow to prove the Geometrization Conjecture in some
cases and outlined a general program of how it could be used to prove the
Geometrization Conjecture in all cases. He ran into problems, however, coping
with certain types of large lumps that tended to grow uncontrollably under
the averaging process.

''What Perelman has done is to figure out
some new and interesting ways to tame these singularities,'' Dr. Mrowka said.
''His work relies heavily on Hamilton's work but makes amazing new contributions
to that program.''

Even if Dr. Perelman's work does not prove the
Geometrization Conjecture, mathematicians said, it is clear that his work
will make a substantial contribution to mathematics.

''This is one
of those happy circumstances where it's going to be fun no matter what,''
Dr. Mrowka said. ''Either he's done it or he's made some really significant
progress, and we're going to learn from it.''